Using TinyCAD XALL feature while keeping faces and their assigned materials
Materials are specifically assigned to faces and faces only, edges can't hold any information about material slots.

If all faces are removed and no renderable geometry is left behind then all material slot assignments will invariably be lost, there is no way to recover material information from edges alone.

This is not a problem specific of TinyCAD or any other addon for that mater, it is just intrinsic to how the data structures work in Blender.

As far as I know there is really no way around this.

You may alternatively try with varying degrees of success using Edit Mode Boolean operations to try and find intersections without loosing the faces; or use tiny cad on a duplicated copy of those edges instead of the original ones attached to faces.